# Equations of motion for charged particles

Igor Khavkine wrote:

Actually, the study of the motion of a point particle coupled to a
classical field has recently been a fairly active research topic. For
example, there is a review article by Eric Poisson published in Living
Reviews in Relativity:

The Motion of Point Particles in Curved Spacetime
by Eric Poisson
http://relativity.livingreviews.org/Articles/lrr-2004-6/

And it seems that, with a slight modification, the Lorentz-Dirac
equation is no longer pathological. The modification is described in

An introduction to the Lorentz-Dirac equation
by Eric Poisson
http://arxiv.org/abs/gr-qc/9912045

The LD equation, as usually stated in a momentarily comoving inertial
frame, is

m a = F + (2/3) q^2 da/dt,

with m - mass of the particle, q - its charge, a - its acceleration,
and F - externally applied force. The above paper argues that this
equation needs to be modified as follows:

m a = F + (2/3) (q^2/m) dF/dt.

As far as I understand, the justification comes from a
renormalization-type argument. The pathology of the original LD
equation (its third order in time derivatives) can be traced to the
point-particle description of the electron. Poisson argues that one
should consider the electron as an extended object and look at its
equations of motion in the limit as its size goes to zero, while
keeping finite mass and charge. At every point of the limiting process,
the equations of motion are second order and free of pathologies. The
result of the limit is precisely the last equation. There, the external
force F is considered to be a vector field on space-time (an
electromagnetic or gravitational field, for instance) and dF/dt
represents its (hydrodynamic) derivative along the particle's
worldline.

These are fairly recent developments. I wonder when they'll make it
into a newer edition of Jackson or some texbook on electrodynamics.

Igor

My guess is that it will be a long time before this equation
is presented as definitive in standard texts.

This equation, which Poisson presents without attribution,
was proposed by Eliezer in 1948:

C. J. Eliezer, "On the classical theory of particles",
Proc. Royal Soc. London A194 (1948), 543-555.

There are good reasons why it has not achieved substantial acceptance
in the succeeding 50-odd years. Some of them are:

(1) No fundamental motivation for it has ever been given.
(This applies also to the motivation in Poisson's recent
paper above.)
It is just one of many *ad hoc* modifications
of the Lorentz-Dirac equation which have been proposed
over the past half century. A survey of some of these
can be found in Section 5.7 of

S. Parrott, "Relativistic electrodynamics and
Differential Geometry", Springer-Verlag, 1987.

That survey unfortunately does not mention the Landau-Lifschitz
equation, another equation of this genre which
is currently advocated by H. Spohn and F. Rohrlich,
among others, e.g.,

H. Spohn, "The Critical Manifold of the Lorentz-Dirac
Equation", Europhys. Lett. 50 (2000), 287-292.

Some criticisms of Spohn's proposals
(which arose in another context)
can be found in Appendix 3 of www.arxiv.org/gr-qc/0502029.

(2) There is no known proof that the Eliezer equation above
conserves energy-momentum.
Since it is only an approximation to the Lorentz-Dirac
equation (which does conserve energy-momentum
under appropriate hypotheses),
it seems unlikely that it does.
So far as I know, no one even suggests that it might.

If an equation of motion does not conserve energy-momentum,
then either that equation is not fundamental,
or we have to revise our basic notions about energy-momentum
(e.g., use an electromagnetic energy-momentum tensor
other than the usual one). Since the usual e-m tensor
is so well established, it would take a brave textbook author
to suggest this.

(3) Most of the proposed second-order approximations to the
Lorentz-Dirac equation, including the Eliezer and
Landau-Lifschitz equations, are free from runaway
or preaccelerative solutions (which are commonly mentioned
pathologies of the Lorentz-Dirac equation).
However, that does not imply that they are free from
*all* pathologies.

For example, the Landau-Lifshitz equation has been criticized
because it implies no radiation reaction (and hence presumably
no radiation) for one-dimensional motion in a constant
electric field, e.g.,

G. Ares de Parga, R. Mares, and S. Dominguez,
"Landau-Lifschitz equation of motion for
a charged particle revisited",
Ann. Found. L. de Broglie, to appear.

So-called *bremsstrahlung" ("braking radiation") is observed
when charged particles are abruptly decelerated
(as when a particle beam smashes into a target).
Although this has probably never been observed in a *constant*
electric field owing to the smallness of the predicted effect,
some physicists think that *bremsstrahlung* should occur when
charged particles are injected into a repulsive constant
electric field.

The Landau-Lifschitz equation implies that *bremsstrahlung*
would *not* be observed in this situation.
So does the Eliezer equation.

The purpose of an equation of motion is to be able to *predict*
the motion of particles in unfamiliar circumstances.
How many textbook authors would be willing to put their
professional reputations on the line by stating that
*bremsstrahlung* in a constant electric field will *never*
be observed? Our present understanding of the effects of radiation
on charged particle motion is too incomplete to make such statements
with confidence.

.

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